A Review of Soil Constitutive Models for Simulating Dynamic Soil–Structure Interaction Processes Under Impact Loading

Abstract

The accurate modeling of dynamic soil–structure interaction processes under impact loading is critical for advancing the design of soil-embedded barrier systems. Full-scale crash testing remains the benchmark for evaluating barrier performance; however, such tests are costly, logistically demanding, and subject to variability that limits repeatability. Recent advancements in computational methods, particularly the development of large-deformation numerical schemes, such as the multi-material arbitrary Lagrangian–Eulerian (MM-ALE) and smoothed particle hydrodynamics (SPH) approaches, offer viable alternatives for simulating soil behavior under impact loading. These methods have enabled a more realistic representation of granular soil dynamics, particularly that of the Manual for Assessing Safety Hardware (MASH) strong soil, a well-graded gravelly soil commonly used in crash testing of soil-embedded barriers and safety features. This soil exhibits complex mechanical responses governed by inter-particle friction, dilatancy, confining pressure, and moisture content. Nonetheless, the predictive fidelity of these simulations is governed by the selection and implementation of soil constitutive models, which must capture the nonlinear, dilatant, and pressure-sensitive behavior of granular materials under high strain rate loading. This review critically examines the theoretical foundations and practical applications of a range of soil constitutive models embedded in the LS-DYNA hydrocode, including elastic, elastoplastic, elasto-viscoplastic, and multi-yield surface formulations. Emphasis is placed on the unique behaviors of MASH strong soil, such as confining-pressure dependence, limited elastic range, and strong dilatancy, which must be accurately represented to model the soil’s transition between solid-like and fluid-like states during impact loading. This paper addresses existing gaps in the literature by offering a structured basis for selecting and evaluating constitutive models in simulations of high-energy vehicular impact events involving soil–structure systems. This framework supports researchers working to improve the numerical analysis of impact-induced responses in soil-embedded structural systems.

1. Introduction

1.1. Numerical Methods for Simulating Dynamic Impact Soil-Structure Interaction Processes

Many soil–foundation interaction problems, such as dynamic pile–soil interaction under impact loading, soil–tool interaction, underground excavation, and dynamic pile driving, involve large and localized deformation of soil materials. These problems are characterized by nonlinear material behavior, extreme loading conditions, and evolving boundary conditions. Despite significant advances in computational methods, accurately modeling these processes remains a challenging task in geotechnical and geomechanical applications. The Updated Lagrangian Finite Element Method (UL-FEM) has been extensively applied to dynamic soil–structure interaction problems over the past several decades. This scheme is based upon fixed mass elements, which means that when the material or structure deforms, so does the attached mesh or grid. Consequently, the mesh becomes severely distorted when the structure or material is subjected to large deformation. Severe mesh distortion inverts the elements, i.e., causes a negative Jacobian or negative volume, which introduces numerical errors or instabilities, slows down the calculation, and sometimes terminates the computation. This issue is particularly evident in situations such as the simulation of rigid or short piles embedded in soil subjected to impact loading, where mesh quality deteriorates rapidly and renders the analysis intractable [1].

The need for more robust approaches to model large deformation problems has driven the development of alternative computational methods grounded in both discrete and continuum theories. Discrete methods, such as the Discrete Element Method (DEM), represent soils as assemblies of individual particles or grains. Continuum-based approaches, which avoid the limitations of traditional mesh-based formulations, include both enhanced mesh-based methods and mesh-free particle methods. These can be broadly categorized into three groups: (1) mesh-based methods incorporating an element erosion algorithm [2,3,4,5,6,7,8,9,10,11,12] or remeshing techniques, such as Eulerian, Simplified Arbitrary Lagrangian–Eulerian (SALE), and Multi-Material Arbitrary Lagrangian–Eulerian (MM-ALE) [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]; (2) mesh-free particle methods, such as the Material Point Method (MPM) [32,33,34], Smoothed Particle Hydrodynamics (SPH) method [34,35,36,37], Particle Finite Element Method (PFEM) [38,39] and many others; and (3) hybrid or adaptive coupling techniques that integrate mesh-based and particle-based approaches [40,41,42,43,44,45,46,47,48,49,50]. A schematic illustrating the computational modeling process for large soil deformation problems is presented in Figure 1.

With the evolution of these large deformation computational methods, it has become feasible to simulate and analyze the complex soil behaviors associated with dynamic impact soil–structure interaction processes [51,52,53]. The accuracy of such simulations is highly dependent on the constitutive model employed to describe the stress–strain response of the soil. In dynamic impact modeling, the constitutive model serves as the fundamental link between stress and strain in each soil element. Given that soil typically exhibits the lowest stiffness and strength within the soil–structure system, its behavior significantly influences the system’s dynamic response. Therefore, it is essential to employ a constitutive model that accurately represents soil behavior over the full range of expected loading conditions.

Over the past five decades, considerable progress has been made in developing and refining soil constitutive models. While early models were relatively simple, advancements in the theoretical understanding of soil mechanics have led to increasingly sophisticated models based on the principles of continuum mechanics. Accurately modeling the behavior of granular soils remains particularly challenging due to several intrinsic characteristics: (1) their stress–strain response is inherently nonlinear; (2) they are frictional materials; and (3) they exhibit dilatancy or contractancy, meaning that volumetric changes occur during shear deformation.

Although continued progress in model development is anticipated, the primary features and theoretical foundations of current soil constitutive models for dynamic soil–structure interaction under high strain rate loading, such as vehicular impact, are reviewed in this paper. This theoretical review focuses on soil models available in LS-DYNA and commonly used in simulations of soil-embedded vehicle barrier systems. Rather than providing an exhaustive treatment of each model, this review aims to clarify their fundamental characteristics and evaluate their suitability for simulating dynamic soil behavior in crash test scenarios.

1.2. Soils in Crash Testing of Barrier Systems

This study focuses on the review of LS-DYNA constitutive models for simulating the dynamic impact response of granular soils, with particular attention to the standard soil used in roadside safety hardware crash testing, as defined by the American Association of State Highway Transportation Officials (AASHTO) Manual for Assessing Safety Hardware 2016 (MASH) [54]. Commonly referred to as MASH strong soil within the roadside safety research community, this material is a well-graded, crushed limestone road base in which gravel-sized particles constitute the dominant grain size fraction. The standard MASH strong soil used in crash tests conducted at the Midwest Roadside Safety Facility (MwRSF) at the University of Nebraska–Lincoln (UNL) conforms to AASHTO specification M 147, grades A or B, which governs materials for aggregate and soil–aggregate subbase, base, and surface courses. Unlike cohesive soils such as clays and silts, the mechanical behavior of granular soils under impact loading, including MASH strong soil, is primarily controlled by nonlinear, short-range, grain-to-grain interactions involving frictional contact and particle collision. These microscale interactions give rise to macroscopic behaviors that are distinct from other geotechnical materials and contribute to the complexity of modeling granular soil responses under high-rate loading conditions.

MASH strong soil exhibits several unusual and highly nonlinear mechanical characteristics that distinguish it from other granular materials. These include (1) strength and stiffness that are highly sensitive to confining pressure, which is influenced by boundary conditions, embedment depth, or experimental setup; (2) dependence on mechanical response on the initial packing density, often quantified by the void ratio or packing fraction and the spatial arrangement of grains; (3) volumetric expansion, or dilatancy, during shear loading; (4) strong sensitivity of stiffness and strength to pore water pressure or moisture content; and (5) a narrow elastic regime, the stress range over which granular soils behave elastically is exceedingly narrow, typically limited to deviatoric or shear stresses below approximately 5–10 kPa and strain levels below 10⁻⁴, beyond which particle rearrangement and irreversible deformations become significant [55,56]. These behaviors, while emerging from relatively simple inter-particle mechanisms, produce complex and sometimes counterintuitive bulk properties that challenge numerical representation.

In practical terms, MASH strong soil can be described as a large assemblage of discrete particles. It is widely utilized in geotechnical engineering applications, including as structural fill behind retaining walls, and is commonly handled across a range of construction practices. Despite its broad use and seemingly straightforward definition, a comprehensive understanding of the dynamic mechanical behavior of this material under impact loading remains limited [42,43,44,45,46]. The difficulty stems from the granular nature of the soil and the dynamic, nonlinear, and dissipative responses that occur under high strain rate excitations such as those imposed during vehicular impacts.

The mechanical response of MASH strong soil during impact loading is governed by interactions between neighboring particles. These interactions, involving both friction and rapid collision, govern the redistribution and dissipation of energy throughout the material. The soil used in most crash testing of soil-based barrier systems and soil-embedded support structures, such as signs and luminaires, is typically well-drained and maintained at a relatively low moisture content to ensure consistency in performance evaluation [54,55,56,57,58,59,60,61]. The grain size distribution of MASH strong soil spans a wide range, from coarse gravel-sized particles to fine silt, as illustrated in Figure 2. Due to the interactions among its constituent particles, MASH strong soil exhibits complex mechanical behavior that reflects characteristics typically associated with multiple states of matter.

As shown in Figure 3, its response differs significantly from that of conventional engineering materials. Under static loading conditions, it may be treated as a bulk solid with relatively stable mechanical properties. In contrast, under dynamic impact conditions, its motion and deformation are often governed by mechanisms that resemble fluid behavior, where inter-particle collisions and momentum exchange dominate. This diverse phenomenological behavior makes MASH strong soil a particularly compelling subject of study. Its ability to transition between solid-like and fluid-like responses presents unique challenges and opportunities for researchers and engineers. A deeper understanding of this behavior is essential for improving crash testing methodologies and for advancing numerical modeling techniques applied to soil-embedded roadside safety systems.

1.3. Scope, Objectives, and Contribution of the Study

The selection of a soil constitutive model for simulating dynamic soil–structure interaction is a decision that rests with the numerical analyst. However, this decision requires a technical understanding of the features that characterize soil behavior under impact loading, and of whether a given model is capable of representing those features in the context of numerical analysis. This paper examines soil constitutive models available in the LS-DYNA hydrocode that have been used in geotechnical simulations involving soil-embedded barrier and containment systems. The models considered include elastic, simple elastoplastic, advanced elastoplastic, elasto-viscoplastic, and multi-surface or nested formulations.

While these models are widely implemented in practice, there has been little work devoted to their theoretical evaluation for problems involving high frequency dynamic loading, such as vehicular impacts. No prior studies have systematically examined the applicability, limitations, and underlying assumptions of these models within the framework of crash test simulations involving dynamic soil–structure interaction processes. A careful examination of these models is necessary to clarify how well each one captures the dominant features of granular soil behavior under dynamic loading. This study reviews the theoretical basis of selected LS-DYNA soil models and evaluates their capabilities and limitations when applied to problems involving soil-embedded systems subjected to impact loading. The intent is to support a more informed approach to the selection and application of constitutive models in computational analyses of soil response during crash testing.

2. Elastic Models

Elastic models were among the earliest tools employed in geotechnical engineering, particularly for estimating foundation settlement using classical approaches, such as Boussinesq theory. The theory of elasticity provides a robust framework for analytical solutions to a range of boundary value problems and continues to offer first-order approximations for the deformation and stability of geotechnical systems where loading remains well below failure thresholds.

Within the LS DYNA simulation environment, two elastic models applicable to granular soils, including MASH strong soil, are available: the Elastic model and the Biot Hysteretic model. These represent isotropic hypoelastic and viscoelastic material formulations, respectively. While both are mathematically tractable and often used for solid granular elements, they exhibit important limitations when applied to dynamic soil–foundation interaction processes under vehicular impact loading.

Elastic models neglect critical aspects of granular soil behavior, including yielding, strain rate dependence, and stress-dependent stiffness and strength. They also fail to capture volume change associated with dilatancy or contractancy during shear loading. Furthermore, experimental studies have demonstrated that the stress range over which granular soils behave elastically is exceedingly narrow [63,64,65]. Consequently, the applicability of these models in impact simulations is limited. A brief theoretical overview of each model is provided below.

2.1. Hypoelastic Model

This material model implements an isotropic hypoelastic formulation, which defines the stress increment as a function of current stress and strain increments. The term “hypoelastic” refers to its incremental nature, indicating that elasticity is defined in a lower-order or differential sense. This allows for path-dependent behavior and makes the model suitable, to a limited extent, for approximating aspects of hardening or softening in granular materials. [64,65,66,67,68]. In the following, the general aspects of the hypoelastic soil model are considered.

The general form of a constitutive relation is given as follows:

$$f\left(\sigma ,\dot{\sigma },\epsilon ,\dot{\epsilon }\right)=0$$

(1)

where and are the stress and stress rate, and and are the strain and strain rate, respectively.

A particular form of Equation (1) is given as follows:

$${\dot{\sigma }}_{ij}=f_{ij}\left({\sigma }_{kl},{\dot{\epsilon }}_{kl}\right)$$

(2)

In hypoelastic models, the response functions are not time-dependent and depend only on the stress tensor:

$${\dot{\sigma }}_{ij}=C_{ijkl}{\dot{\epsilon }}_{kl}$$

(3)

For isotropic hypoelasic granular soil, this becomes the following:

$$\begin{gathered} {\dot{\sigma }}_{ij}=a_0{\dot{\epsilon }}_{kk}{\delta }_{ij}+a_1{\dot{\epsilon }}_{ij}+a_2{\dot{\epsilon }}_{kk}{\sigma }_{ij}+a_3{\sigma }_{mn}{\dot{\epsilon }}_{mn}{\delta }_{ij}+a_4\left({\sigma }_{im}{\dot{\epsilon }}_{mj}+{\dot{\epsilon }}_{im}{\sigma }_{mj}\right) \\ +a_5{\dot{\epsilon }}_{kk}{\sigma }_{im}{\sigma }_{mj}+a_6{\sigma }_{mn}{\dot{\epsilon }}_{nm}{\sigma }_{ij}+a_7{\sigma }_{mn}{\sigma }_{nk}{\dot{\epsilon }}_{km}{\delta }_{ij} \\ +a_8\left({\sigma }_{im}{\sigma }_{mk}{\dot{\epsilon }}_{kj}+{\dot{\epsilon }}_{im}{\sigma }_{mk}{\sigma }_{kj}\right)+a_9{\sigma }_{mn}{\dot{\epsilon }}_{nm}{\sigma }_{ik}{\sigma }_{kj} \\ +a_{10}{\sigma }_{mn}{\sigma }_{nk}{\dot{\epsilon }}_{km}{\sigma }_{ij}+a_{11}{\sigma }_{mn}{\sigma }_{nk}{\dot{\epsilon }}_{km}{\sigma }_{ir}{\sigma }_{rj} \end{gathered}$$

(4)

The coefficients $a_0,a_1,a_2,..............,a_{11}$ are functions of stress invariants. This general formulation is widely discussed in the literature in hypoelasticity and has been used to derive specific forms for different hypoplastic grades [66,67,68,69,70] to obtain special forms of various grades or orders of the hypoelastic model. For example, in a grade-zero hypoelastic model, the constitutive equation is assumed to be independent of stress; the simplified form is as follows:

$${\dot{\sigma }}_{ij}=a_0{\dot{\epsilon }}_{kk}{\delta }_{ij}+a_1{\dot{\epsilon }}_{ij}$$

(5)

with $a_0=\lambda$ and $a_1=\mu$, where $\lambda$ and $\mu$ are Lame’s constants. Equation (5) corresponds to the generalized Hooke’s law in the following rate form:

$${\dot{\sigma }}_{ij}=\left(K-{\displaystyle \frac{2G}{3}}\right){\dot{\epsilon }}_{kk}{\delta }_{ij}+2G{\dot{\epsilon }}_{ij}$$

(6)

This relation can also be expressed in incremental form as follows:

$$d{\sigma }_{ij}=\left(K-{\displaystyle \frac{2G}{3}}\right)d{\epsilon }_{kk}{\delta }_{ij}+2Gd{\epsilon }_{ij}$$

(7)

The theoretical basis and implementation of hypoelastic soil models, particularly in the context of granular and cohesionless materials, have been explored in several seminal works. These include the early formulations by Truesdell [66,67], studies on recoverable deformation in cohesionless soils by Coon and Evans [68], the finite element implementation of hypoelasticity by Mysore [69], and investigations of elastic behavior in cohesionless soils by Holubec [70].

2.2. Biot Hysteretic Model

In his foundational paper titled Linear Thermodynamics and the Mechanics of Solids, Biot [71], proposed a viscoelastic model that describes hysteretic behavior as the limiting case of a general viscoelastic formulation. This model combines a linear elastic (Hookean) element in parallel with an infinite number of Maxwell elements, each defined by a specific stiffness and relaxation time [72]. The Biot hysteretic model is a linear viscoelastic formulation that can simulate the nearly frequency-independent response of granular materials subjected to cyclic or earthquake loading.

The implementation of this model in LS-DYNA is referred to as the Biot Hysteretic model. It enables time-domain viscoelastic analysis for materials, such as granular soils [73]. In this formulation, stress is expressed as a convolution integral of the strain rate with a relaxation modulus:

$${\sigma }_{ij}\left(t\right)={{\displaystyle \int }}_0^t{C}_R\left(t-\xi \right){\dot{\epsilon }}_{ij}\left(\xi \right)d\xi$$

(8)

where ${\sigma }_{ij}\left(t\right)$ is the stress at the current time? $t$, ${\dot{\epsilon }}_{ij}$ is t strain rate history, and $C_R\left(t\right)$ is the relaxation modulus. The relaxation modulus describes the stress response at time $t$ due to a unit step strain applied at time $\xi$ where $\xi \left(\xi \le t\right)$.

The relaxation modulus is defined as follows:

$$C_R\left(t\right)=C_1\left[1+{\displaystyle \frac{2\eta }{\pi }}{E}_i\left(-\beta t\right)\right]$$

(9)

In this expression, $\eta ={\displaystyle \frac{C_2}{C_1}}$ is the hysteretic damping coefficient, $\beta ={\displaystyle \frac{2\pi F_D}{10}}$ where $F_D$ is the dominant excitation frequency in Hz, and $E_i\left(x\right)$ is the exponential integral given by [73]:

$$E_i\left(x\right)={{\displaystyle \int }}_X^{\infty }{\displaystyle \frac{e^{-\xi }}{\xi }}d\xi$$

(10)

A schematic representation of Biot’s hysteretic model is shown in Figure 4. It consists of a spring with stiffness $C_1$ in parallel with an infinite number of Maxwell elements, each represented by a spring and dashpot stiffness $K_i$, and relaxation time ${\mu }_i$.

2.3. Advantages and Limitations of Elastic Soil Models

The advantages and limitations of elastic models implemented in LS-DYNA, namely Hypoelastic model and Biot Hysteretic model are presented in

Table 1. Advantages and limitations of LS-DYNA elastic soil models.

Model	Advantages	Limitations
Elastic	Conceptually and mathematically simple	Does not include coupling between deviatoric and volumetric responses, so it cannot simulate dilation during shear deformation.
	Requires few material parameters and is easy to calibrate	Behavior is incrementally reversible and does not capture yielding or stress-dependent stiffness and strength.
	Supported by a wide range of experimental data	Inadequate for modeling behavior near failure, and cannot simulate strain softening or strain rate effects
	Have been used successfully in many practical applications	Cannot simulate strain-softening as well as strain rate effects
	Commonly used in practice for preliminary assessments	Does not account for low confinement conditions, which are important in roadside barrier systems involving shallow soil fill
Biot Hysteretic	Capable of simulating hysteretic energy dissipation under cyclic or dynamic loading	Cannot represent strain rate effects or strain softening.
		Cannot simulate strain rate as well as strain softening effects.
		Does not capture yielding or stress-dependent stiffness
		Neglects low confinement effects, which are essential in the context of roadside barrier systems embedded in granular soil

. These models are sometimes used in the computational analysis of dynamic soil–structure interaction problems, including large deformation simulations of pile–soil systems. While both models are relatively easy to implement and require a limited number of input parameters, they are not well suited for problems involving nonlinear or strain rate-dependent behavior, particularly under impact loading conditions.

3. Simple Elastoplastic Models

Simple elastoplastic constitutive models are frequently employed in LS-DYNA simulations to represent soil behavior under blast loading, tire–soil interaction, and quasi-static soil–foundation systems [73,74,75,76,77,78,79]. A theoretical overview of each model is presented below, followed by a discussion of its advantages and limitations in modeling dynamic soil–structure interaction processes.

3.1. Soil and Foam Model

The Soil and Foam model is a pressure-sensitive, elastoplastic formulation initially developed by Kraig [80] to simulate the behavior of cellular concrete. It is one of the simplest soil models available in LS-DYNA. The model incorporates a yield surface defined in terms of mean and deviatoric stress, allows for elastic unloading, and includes a tensile cut-off criterion. The yield surface has a parabolic form capped by a pressure-dependent ellipsoidal boundary. The stress response is elastic at small strains, while nonlinear behavior due to void collapse and plastic deformation appears at larger strains. Shear strength increases with confining pressure due to cohesion and frictional effects.

The yield function is defined in terms of the second invariant of deviatoric stress $J_2={\displaystyle \frac{1}{2}}{s}_{ij}{s}_{ij}$ and mean stress $p$, expressed as follows:

$$\varphi =\left(p-f\right)\left[J_2-\left(a_0+a_{1}p+a_2{p}^2\right)\right]=0$$

(11)

where $a_0$, $a_1$, and $a_2$ are material constants and $f$ is a function of the mean total strain. The deviator stress is as follows:

$$s_{ij}={\sigma }_{ij}+p{\delta }_{ij}$$

(12)

in which ${\delta }_{ij}$ is the Kronecker delta, and the pressure is linked to the stress trace as follows:

$$p=-{\displaystyle \frac{1}{3}}{\sigma }_{kk}$$

(13)

The theoretical formulation becomes more tractable by introducing two scalar functions, one describing the deviatoric part of the yield surface and the other representing a plane orthogonal to the hydrostatic axis [80]:

$${\varphi }_s=J_2-\left(a_0+a_{1}p+a_2{p}^2\right)$$

(14)

$${\varphi }_p=p-f$$

(15)

The total strain rate ${\dot{\epsilon }}_{ij}$ is separated into volumetric strain rate, ${\dot{e}}_{ij}$ and the deviatoric strain rate, ${\dot{E}}_{ij}$, as follows:

$${\dot{e}}_{ij}={\displaystyle \frac{1}{3}}{\dot{\epsilon }}_{kk}{\delta }_{ij}$$

(16)

$${\dot{E}}_{ij}={\dot{\epsilon }}_{ij}-{\displaystyle \frac{1}{3}}{\dot{\epsilon }}_{kk}{\delta }_{ij}$$

(17)

Plastic volumetric behavior is determined by evaluating whether the stress state lies on the yield surface and whether the loading condition is active:

$${\varphi }_p=0 \mathrm{and} {\dot{\varphi }}_p=0$$

(18)

If these conditions are not met, the material behaves elastically in the volumetric mode:

$$\dot{p}=-3K{\dot{e}}_{ij}$$

(19)

where $K$ is the bulk modulus. Similarly, the deviatoric response is elastic if ${\varphi }_s<0$ or ${\varphi }_s=0$ and ${\dot{\varphi }}_s<0$.

In this case, the stress increment is as follows:

$${\dot{s}}_{ij}=2G{\dot{\epsilon }}_{ij}$$

(20)

where $G$ is the shear modulus. For loading on the yield surface,

$${\varphi }_s=0 \mathrm{and} {\displaystyle \frac{\partial {\varphi }_s}{\partial s_{ij}}}{\dot{s}}_{ij}\ge 0$$

(21)

a plastic correction is required. The elastic deviatoric strain rate is

$${\dot{E}}^{e}ij={\displaystyle \frac{{\dot{s}}_{ij}}{2G}}$$

(22)

and the plastic deviatoric strain rate is given as follows:

$${\dot{E}}^{p}ij={\dot{\epsilon }}_{ij}-{\dot{E}}_{ij}^e$$

(23)

The model assumes associative flow in deviatoric stress space:

$${\dot{E}}^{p}ij=\lambda {\displaystyle \frac{\partial {\varphi }_s}{\partial s_{ij}}}$$

(24)

where $\lambda$ is a plastic multiplier determined from the consistency condition:

$${\dot{\varphi }}_s={\displaystyle \frac{\partial {\varphi }_s}{\partial s_{ij}}}{\dot{s}}_{ij}+{\displaystyle \frac{\partial {\varphi }_s}{\partial p}}\dot{p}=0$$

(25)

Given ${\varphi }_s$ depends on $J_2$ is, its gradient is

$${\displaystyle \frac{\partial {\varphi }_s}{\partial s_{ij}}}={\displaystyle \frac{\partial J_2}{\partial s_{ij}}}=s_{ij}$$

(26)

which leads to the following relation between stress rate and pressure rate:

$$s_{ij}{\dot{s}}_{ij}=\left(a_1+2a_{2}p\right)\dot{p}$$

(27)

An expression for ${\dot{s}}_{ij }$is derived as

$${\dot{s}}_{ij}=2G\left({\epsilon }_{ij}-\lambda s_{ij}\right)$$

(28)

and the plastic multiplier is

$$\lambda ={\displaystyle \frac{s_{ij}{\dot{E}}_{ij}^p-\left(a_1+2a_{2}p\right)\frac{\dot{p}}{2G}}{s_{ij}{s}_{ij}}}$$

(29)

Substitution into Equation (28) provides the stress rate update:

$${\dot{s}}_{ij}=2G{\dot{\epsilon }}_{ij}-\left[{\displaystyle \frac{2G{s}_{ij}{\epsilon }_{ij}-\left(a_1+2a_{2}p\right)\dot{p}}{s_{ij}{s}_{ij}}}\right]s_{ij}$$

(30)

Since the deviatoric plastic strain increment is normal to the yield surface, the model does not predict volumetric plastic strain during shear. This restriction fails to capture dilation, which is frequently observed in dense granular soils under dynamic shear due to the disruption of interparticle arrangements. This limitation is critical for simulations of dynamic soil–structure interaction processes where granular fill exhibits volumetric expansion under impact.

3.2. Soil and Foam Failure Model

The Soil and Foam Failure model shares the same constitutive structure as Soil and Foam model. The key distinction is the inclusion of a failure pressure parameter that defines the tensile strength limit. Once this pressure is exceeded, the soil element loses tensile load-carrying capacity. The theoretical formulation remains unchanged from the preceding model.

3.3. Pseudo-Tensor Model

The Pseudo-Tensor model is one of the earliest constitutive formulations in LS-DYNA [73]. It was originally developed to simulate reinforced concrete, but can be adapted to granular materials by disabling reinforcement-related parameters [77]. The model defines independent surfaces for pressure-dependent compaction and shear failure. Two response modes are supported. In mode I, the user defines the pressure-dependent strength behavior of the soil through tabulated data, typically representing the deviatoric stress (stress difference) as a function of mean pressure, thereby enabling accurate calibration to experimental triaxial test results. Mode II allows for the definition of two shear strength curves and includes a damage parameter for progressive failure. Mode II is especially useful when only unconfined compressive strength is available. Reinforcing bars can also be modeled in a smeared representation. Despite this flexibility, the model is not appropriate for simulating unreinforced granular soils under low-confinement impact conditions. Therefore, it is excluded from further detailed theoretical discussion.

3.4. Advantages and Limitations of Simple Elastoplastic Soil Models

The advantages and limitations of the previously discussed simple elastoplastic models, i.e., Soil and Foam, Soil and Foam Failure, and Pseudo-Tensor for large deformation computational modeling of dynamic soil–structure interactions processes, are summarized in

Table 2. Advantages and limitations of simple elastoplastic models.

Model	Advantages	Limitations
Soil and Foam and Soil and Foam Failure	Relatively simple to implement when tabulated pressure versus volumetric strain data is available	Requires confinement to maintain numerical stability; developed primarily for high-pressure condition
	Material parameters can be calibrated using Mohr–Coulomb cohesion and friction angle	Unable to capture volumetric expansion (dilation) of granular soil during shear under impact loading
		Does not account for pore pressure build-up or moisture effects during undrained shear failure
		Inadequate for modeling strain-softening behavior and strain rate effects
		Assumes a circular yield surface in the deviatoric plane, which is inconsistent with experimental observations for granular soils
Pseudo Tensor	Straightforward to use when yield stress versus pressure data is available in tabulated form	Neglects low-confinement effects, which are significant in roadside safety applications
	Capable of simulating strain softening and rate-dependent behavior	Cannot represent pore pressure build-up or moisture-related effects in undrained dynamic loading
		Fails to capture dilative volumetric behavior of soil during impact-driven shear

.

4. Elastoplastic Models

Soil constitutive models in this category include Soil Concrete, Mohr–Coulomb, Drucker–Prager, and Jointed Rock. These models are widely implemented in commercially available finite element software and are commonly applied in geotechnical foundation design. A theoretical overview of their elastoplastic frameworks is presented below.

4.1. General Elastoplasticity Framework

The foundational assumption of elastoplastic soil models is that the strain and strain rates decompose into elastic and plastic parts:

$${\dot{\epsilon }}_{ij}={\dot{\epsilon }}_{ij}^{\left(el\right)}+{\dot{\epsilon }}_{ij}^{\left(pl\right)}$$

(31)

where ${\dot{\epsilon }}_{ij}$ is the total strain rate tensor, ${\dot{\epsilon }}_{ij}^{\left(el\right)}$ is the elastic part, and ${\dot{\epsilon }}_{ij}^{\left(pl\right)}$ is the plastic part.

The elastic strain rate follows Hooke’s law:

$${\dot{\epsilon }}_{ij}^{\left(el\right)}={\displaystyle \frac{{\dot{s}}_{ij}}{2G}}+{\displaystyle \frac{1-2\nu }{3E}}{\dot{\sigma }}_{kk}{\delta }_{ij}$$

(32)

where ${\dot{s}}_{ij}$ is the deviator stress rate tensor, $G$ is shear modulus, $\nu$ is Poison’s ratio, $E$ is Young’s modulus, and ${\delta }_{ij}$ is the Kronecker delta tensor. The term ${\dot{\sigma }}_{kk}={\dot{\sigma }}_{xx}+{\dot{\sigma }}_{yy}+{\dot{\sigma }}_{zz}$. For simplicity, elastic constants $G$, $E$, and $\nu$ are treated as constant. Although they are known to vary with stress state in granular soils [81,82,83,84,85].

Following Hill [86], the plastic strain rates are proportional to the derivative of the plastic potential function $g$ with respect to stresses:

$${\dot{\epsilon }}_{ij}^{\left(pl\right)}=\dot{\lambda }{\displaystyle \frac{\partial g}{\partial {\sigma }_{ij}}}$$

(33)

Here, $\dot{\lambda }$ is the rate form of the plastic multiplier, which is positive in plastic flow and zero in elastic behavior. The plastic potential function defines the direction of plastic strain increments and may or may not coincide with the yield function $f$. The associate flow rule implies $g=f$; the non-associated rule $g$ Ē$f$.

The plastic multiplier satisfies the conditions:

$$\dot{\lambda }=0 \mathrm{for} f=0 \mathrm{or} f<0 \mathrm{and} df<0 \left(\mathrm{elastic}\mathrm{or}\mathrm{plastic}\mathrm{unloading}\right)$$

(34)

$$\dot{\lambda }<0 \mathrm{for} f=0 \mathrm{and} df=0 \left(\mathrm{plastic}\mathrm{loading}\right)$$

(35)

The Prager consistency condition enforces that the stress state remains on the yield surface during plastic flow:

$$df={\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}d{\sigma }_{ij}=0$$

(36)

This guarantees that the new stress state ${\sigma }_{ij}+d{\sigma }_{ij}$ after loading fulfills the yield criterion. Thus, a material point remains in the plastic state. The consistency condition is expressed as follows:

$$f\left({\sigma }_{ij}+d{\sigma }_{ij}\right)=f\left({\sigma }_{ij}\right)+df=f\left({\sigma }_{ij}\right)$$

(37)

Substituting Equations (31) and (32) into the total strain rate provides the following:

$${\dot{\epsilon }}_{ij}={\displaystyle \frac{{\dot{s}}_{ij}}{2G}}+{\displaystyle \frac{1-2\nu }{E}}{\dot{\sigma }}_{kk}{\delta }_{ij}+\dot{\lambda }{\displaystyle \frac{\partial g}{\partial {\sigma }_{ij}}}$$

(38)

The total stress tensor is written as follows:

$${\sigma }_{ij}=s_{ij}+{\displaystyle \frac{1}{3}}{\sigma }_{kk}{\delta }_{ij}$$

(39)

Substituting this into the governing formulation yields the general elastoplastic stress–strain relation:

$${\dot{\sigma }}_{ij}=2G{\dot{e}}_{ij}+K{\dot{\epsilon }}_{kk}{\delta }_{ij}-\dot{\lambda }\left[\left(K-{\displaystyle \frac{2G}{3}}\right){\displaystyle \frac{\partial g}{\partial {\sigma }_{pq}}}{\delta }_{pq}{\delta }_{ij}+2G{\displaystyle \frac{\partial g}{\partial {\sigma }_{ij}}}\right]$$

(40)

in which $i$ and $j$ are free indices; $p$ and $q$ are dummy indices, ${\dot{e}}_{ij}={\dot{\epsilon }}_{ij}-{\displaystyle \frac{1}{3}}{\dot{\epsilon }}_{ij}{\delta }_{ij}$ is a deviator shear strain rate, and $K$ is the bulk modulus.

The bulk and shear moduli are related to the elastic modulus as follows:

$$G={\displaystyle \frac{E}{2\left(1+\nu \right)}} \mathrm{and} K={\displaystyle \frac{E}{3\left(1-2\nu \right)}}$$

(41)

The rate of the plastic multiplier is obtained from:

$$\dot{\lambda }={\displaystyle \frac{2G{\dot{\epsilon }}_{ij}\frac{\partial f}{\partial {\sigma }_{ij}}+\left(K-\frac{2G}{3}\right){\dot{\epsilon }}_{kk}\frac{\partial f}{\partial {\sigma }_{ij}}{\delta }_{ij}}{2G\frac{\partial f}{\partial {\sigma }_{pq}}\frac{\partial g}{\partial {\sigma }_{pq}}+\left(K-\frac{2G}{3}\right)\frac{\partial f}{\partial {\sigma }_{pq}}{\delta }_{pq}\frac{\partial g}{\partial {\sigma }_{pq}}{\delta }_{pq}}}$$

(42)

Once $f$ and $g$ are specified, the above framework can fully determine the stress evolution of elastoplastic materials.

4.2. Soil Concrete Model

The Soil Concrete model shares several features with the Soil and Foam model. However, it provides additional capabilities for simulating crack initiation and post-peak softening, including residual strength. The reader is referred to the earlier theoretical description of Soil and Foam for its base framework. This model was primarily developed to capture the behavior of concrete, including tensile cracking and post-cracking stiffness reduction. It is not suitable for modeling dynamic impact responses of granular soils such as MASH strong soil, which are typically encountered in roadside safety applications. For this reason, the Soil Concrete model is not included in the present theoretical evaluation of soil constitutive models applicable to roadside safety systems.

4.3. Mohr–Coulomb

The Mohr–Coulomb model is a non-associated elastoplastic model that uses a Mohr–Coulomb yield surface defined by friction angle and cohesion. The dilation and friction angles can vary as functions of accumulated plastic strain [73]. The Mohr–Coulomb (MC) model is widely used in geotechnical and foundation engineering due to its relatively simple form and minimal input requirements, which can be obtained from routine laboratory testing [87]. In addition to two elastic parameters, Young’s modulus $E$ and poison’s ratio $\nu$ the model requires three parameters for plastic behavior: the friction angle $\varphi$, the cohesion $c$, and the dilation angle $\psi$. The dilation angle is less frequently reported than the friction angle or cohesion, but can be measured using drained triaxial compression testing.

The MC yield criterion generalizes Coulomb’s friction law to three-dimensional stress states. Yield occurs when one of the following is satisfied:

$$\begin{gathered} f_1=\left|{\sigma }_1-{\sigma }_2\right|+\left({\sigma }_1+{\sigma }_2\right)\mathit{sin}\varphi -2c\mathit{cos}\varphi =0 \\ {f}_2=\left|{\sigma }_2-{\sigma }_3\right|+\left({\sigma }_2+{\sigma }_3\right)\mathit{sin}\varphi -2c\mathit{cos}\varphi =0 \\ {f}_3=\left|{\sigma }_1-{\sigma }_3\right|+\left({\sigma }_1+{\sigma }_3\right)\mathit{sin}\varphi -2c\mathit{cos}\varphi =0 \end{gathered}$$

(43)

where ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$ are the principal stresses. These expressions define a yield surface that forms a pyramid in principal stress space. Since soils cannot sustain tensile stresses, the surface is truncated along the planes ${\sigma }_1=0,\hspace{0.33em}{\sigma }_2=0,\mathrm{and} \hspace{0.33em}{\sigma }_3=0$. In the deviatoric plane, the cross-section of the yield surface is a noncircular hexagon, as illustrated in Figure 5.

The non-associated plastic potential functions for each stress pair are as follows:

$$\begin{gathered} g_1=\left|{\sigma }_1-{\sigma }_2\right|+\left({\sigma }_1+{\sigma }_2\right)sin\psi \\ {g}_2=\left|{\sigma }_2-{\sigma }_3\right|+\left({\sigma }_2+{\sigma }_3\right)sin\psi \\ {g}_3=\left|{\sigma }_1-{\sigma }_3\right|+\left({\sigma }_1+{\sigma }_3\right)sin\psi \end{gathered}$$

(44)

in the associated flow case $\psi =\varphi$.

4.4. Drucker Prager Model

The Drucker Prager model is a pressure-sensitive elastoplastic formulation developed by modifying the von Mises yield criterion to incorporate confinement effects [88]. The Drucker–Prager yield function is given as follows:

$$f=\sqrt{J_2}-\alpha I_1-k=0$$

(45)

in which $J_2$ is the second invariant of the deviatoric stress tensor, $I_1$is the first stress invariant, and $\alpha$ and $k$ are material constants related to friction and cohesion, respectively. For purely cohesive materials $\alpha =0$, and the yield criterion reduces to von Mises form. The second invariant of deviator stress and first invariant of stress tensors are defined as follows:

$$J_2={\displaystyle \frac{1}{2}}{s}_{ij}{s}_{ij} \mathrm{and} I_1={\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}$$

(46)

Figure 6 illustrates the yield surface in the principal stress space as a circular cone, and Figure 7 shows its projection in the $\sqrt{J_2}-p$ plane.

When the yield function is not satisfied:

$$f=\sqrt{J_2}-\alpha I_1-k<0, \mathrm{or} f=0 \mathrm{and} df={\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}d{\sigma }_{ij}<0$$

$${\dot{\sigma }}_{ij}=K{\dot{\epsilon }}_{kk}{\delta }_{ij}+2G{\dot{e}}_{ij}$$

(47)

When yielding occurs:

$$f=0 \mathrm{and} df=0$$

$${\dot{\sigma }}_{ij}=K{\dot{\epsilon }}_{kk}{\delta }_{ij}+2G{\dot{e}}_{ij}-\dot{\lambda }\left[-3K\alpha {\delta }_{ij}+{\displaystyle \frac{G{s}_{ij}}{\sqrt{J_2}}}\right]$$

(48)

The rate of the plastic multiplier is computed from the following:

$$\dot{\lambda }={\displaystyle \frac{-3K\alpha {\dot{\epsilon }}_{kk}+\frac{G}{\sqrt{J_2}}{s}_{pp}{\dot{e}}_{pq}}{9K{\alpha }^2+G}}$$

(49)

Material constants, $\alpha$and $k$are related to the friction angle $\varphi$ and cohesive $c$ as follows [89,90,91]:

$$\alpha ={\displaystyle \frac{2\mathit{sin}\varphi }{\sqrt{3}\left(3-\mathit{sin}\varphi \right)}} \mathrm{and} k={\displaystyle \frac{6c\mathit{cos}\varphi }{\sqrt{3}\left(3-\mathit{sin}\varphi \right)}}$$

(50)

Plastic volumetric strain rate is as follows:

$${\dot{\epsilon }}_{kk}^p=-3\alpha \left[{\displaystyle \frac{-3K\alpha {\dot{\epsilon }}_{kk}+\frac{G}{\sqrt{J_2}}{s}_{pq}{\dot{e}}_{pq}}{9K{\alpha }^2+G}}\right]$$

(51)

This expression shows that the model inherently produces volumetric expansion during shear, representing dilation behavior due to scalar nonlinearity [89].

4.5. Jointed Rock Model

The Jointed Rock model is an anisotropic elastoplastic model developed to represent jointed or stratified rock masses. Anisotropy may be elastic, characterized by different stiffness in different directions, or plastic, represented by varying friction angle, cohesion, and dilation angle across planes of weakness. The LS-DYNA implementation does not include kinematic hardening [83].

The model assumes that the rock matrix behaves as a transversely anisotropic elastic material, with up to three orthogonal joint sets that limit shear and tensile strength in specified directions. Each plane may have independent shear strength properties. Tensile stresses normal to the joints are capped by tension cut-off values. Sliding is activated once shear strength is reached in any joint direction.

The Mohr–Coulomb failure criterion limits the shear stresses in the principal joint direction. Thus, plastic sliding will occur when maximum shear stress is reached in the principal joint direction. In the Jointed Rock model, a maximum of three sliding planes or directions can be defined. The first direction is assumed to coincide with the plane of elastic anisotropy, and each direction can have different shear strength properties. Also, a predefined tension cut-off, i.e., tensile strength, limits the tensile stresses perpendicular to the three planes.

This model is applicable when rock masses contain regular, parallel joint sets and no fault gouge. However, it is not suitable for modeling granular soils used in crash testing of soil-embedded roadside safety structures. Therefore, it is excluded from further analysis in the present study. For detailed formulation, the reader is referred to [92,93,94,95].

4.6. Advantages and Limitations of Elastoplastic Soil Models

The Drucker–Prager model overestimates frictional resistance in non-axisymmetric stress states due to its circular deviatoric projection. Both Drucker–Prager and Mohr–Coulomb models require elastic parameters that are difficult to define for granular soils, which exhibit nonlinear inelastic response from the onset of deformation. Additionally, these models cannot represent pore pressure generation during undrained shear under dynamic loading. The main advantages and limitations of the Mohr–Coulomb and Drucker–Prager models for modeling dynamic soil–structure interaction processes are summarized in

Table 3. Advantages and limitations of elastoplastic soil models.

Model	Advantages	Limitations
Drucker–Prager	Simple to use	Exhibits excessive plastic dilation during yielding
	Can be calibrated with Mohr–Coulomb parameters by appropriate selection of constants	Cannot capture hysteretic stress–strain behavior within the yield surface
	Satisfies uniqueness conditions due to associated flow rule	Does not model pore pressure build-up or moisture effects during undrained shear under dynamic loading
	Smooth, continuous yield surface in stress space	Circular deviatoric section disagrees with experimental data for granular soils
		Fails to represent low-confinement effects typical in roadside safety applications
		Does not simulate material damage or strain rate effects
Mohr–Coulomb	Simple to use	May cause numerical instability due to sharp corners in the yield surface
	Its validity is well-established for many granular soils	Neglects influence of intermediate principal stress
		Hexagonal deviatoric section inconsistent with laboratory results
		Does not capture low-confinement behavior characteristic of surface-embedded systems
		Cannot model moisture effects or pore pressure generation in undrained shear under impact conditions

.

5. Elasto-Viscoplastic Model

5.1. FHWA Model

The FHWA soil constitutive model was developed by APTEK, Inc. to simulate the mechanical behavior of granular soils under impact loading in crash tests, particularly those governed by NCHRP Report 350 protocols [96]. This model integrates a regularized Mohr–Coulomb yield surface based on the framework proposed by Abbo and Sloan [97], and incorporates critical features necessary for modeling realistic soil response in roadside safety applications. These features include confinement-dependent strength, rate sensitivity, strain softening, pore pressure evolution, and numerical erosion.

The model is formulated within an elasto-viscoplastic framework. The elastic response follows isotropic linear elasticity, consistent with Hookean behavior typically observed in compacted, well-graded soils. The plastic response is governed by a regularized Mohr–Coulomb yield function, while rate-dependent effects are introduced via a Duvaut–Lions-type viscoplastic regularization scheme.

5.1.1. Elastic Response

The elastic strain rate tensor is governed by classical linear elasticity:

$${\dot{\epsilon }}_{ij}^{\left(el\right)}={\displaystyle \frac{{\dot{s}}_{ij}}{2G}}+{\displaystyle \frac{1-2\nu }{3E}}{\dot{\sigma }}_{kk}{\delta }_{ij}$$

(52)

where ${\dot{s}}_{ij}$ is the deviator stress rate tensor, $G$ is the shear modulus, $\nu$ is the Poisson’s ratio, $E$ is the Young’s modulus, ${\dot{\sigma }}_{kk}={\dot{\sigma }}_{11}+{\dot{\sigma }}_{22}+{\dot{\sigma }}_{33}$, and ${\delta }_{ij}$ is the Kronecker delta.

5.1.2. Yield Condition

The plastic yield condition is a regularized variant of the classical Mohr–Coulomb (MC) criterion, originally formulated to describe frictional failure of granular media. It provides a pressure-dependent shear strength envelope defined in terms of principal stresses as follows:

$$f=\left({\sigma }_1-{\sigma }_3\right)+\left({\sigma }_1+{\sigma }_3\right)\mathit{sin}\varphi -2c\mathit{cos}\varphi =0$$

(53)

where ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$ are the principal stresses (compressive stresses taken as negative), $\varphi$ is the friction angle soil, and $c$is the cohesion.

To circumvent the direct evaluation of principal stresses in three dimensions, the criterion is recast using stress invariants [98]:

$${\sigma }_m={\displaystyle \frac{1}{3}}\left({\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}\right)$$

$$\overline{\sigma }=\sqrt{J_2}=\sqrt{{\displaystyle \frac{1}{2}}\left(s_{11}^2+s_{22}^2+s_{33}^2\right)+{\sigma }_{12}^2+{\sigma }_{13}^2+{\sigma }_{23}^2}$$

$$\theta ={\displaystyle \frac{1}{3}}{\mathit{sin}}^{-1}\left(-{\displaystyle \frac{3\sqrt{3}}{2}}{\displaystyle \frac{J_3}{{\left(J_2\right)}^{\frac{3}{2}}}}\right)-{30}^0\le \theta \le {30}^0$$

In which

$$J_3=s_{11}{s}_{22}{s}_{33}+2{\sigma }_{12}{\sigma }_{23}{\sigma }_{13}-s_{11}{\sigma }_{23}^2-s_{22}{\sigma }_{13}^2-s_{33}{\sigma }_{12}^2$$

and

$$s_{11}={\sigma }_{11}-{\sigma }_m,\hspace{1em}{s}_{22}={\sigma }_{22}-{\sigma }_m,\hspace{1em}{s}_{33}={\sigma }_{33}-{\sigma }_m$$

The principal stresses can be expressed in terms of the stress invariants as follows:

$${\sigma }_1={\displaystyle \frac{2}{\sqrt{3}}}\sqrt{J_2}\mathit{sin}\left(\theta +{120}^0\right)+{\sigma }_m$$

(54)

$${\sigma }_2={\displaystyle \frac{2}{\sqrt{3}}}\sqrt{J_2}\mathit{sin}\left(\theta \right)+{\sigma }_m$$

(55)

$${\sigma }_3={\displaystyle \frac{2}{\sqrt{3}}}\sqrt{J_2}\mathit{sin}\left(\theta -{120}^0\right)+{\sigma }_m$$

(56)

By substituting Equations (54) and (56) into Equation (53), the MC yield becomes:

$$f={\sigma }_m\mathit{sin}\varphi +\sqrt{J_2}K\left(\theta \right)-c\mathit{cos}\varphi =0$$

(57)

where

$$K\left(\theta \right)=\mathit{cos}\theta -{\displaystyle \frac{1}{\sqrt{3}}}\mathit{sin}\varphi \mathit{sin}\theta$$

(58)

The MC yield criterion defines a hexagonal pyramid in three-dimensional principal stress space, with its central axis aligned along the hydrostatic axis [99]. In the octahedral plane, corresponding to constant mean stress, the MC yield surface forms an irregular hexagon characterized by sharp vertices where the surface gradient is discontinuous. These corners represent singularities where the first derivative of the yield function is not defined, i.e., the yield function lacks C¹ continuity. As stress states in geotechnical applications frequently approach these corners, the lack of smoothness becomes a practical and numerical limitation.

To overcome this deficiency, the FHWA soil model adopts a regularization scheme based on the hyperbolic smoothing approach proposed by Abbo and Sloan [97]. For a fixed Lode angle $\theta$, the MC criterion describes a straight line in the $\sqrt{J_2}$- ${\sigma }_m$meridional plane, as shown in Figure 8. This line can be expressed as follows:

$$\sqrt{J_2}={\displaystyle \frac{1}{K\left(\theta \right)}}\left(c\mathit{cos}\varphi -{\sigma }_m\mathit{sin}\varphi \right)$$

(59)

where the slope is $-\mathit{sin}\varphi /K\left(\theta \right)$, and the line intersects ${\sigma }_m$-axis at ${\sigma }_m=c\cot \varphi$. To obtain a differentiable surface, Abbo and Sloan introduced a hyperbolic approximation with asymptotes matching the MC linear yield line [100]. The general form of this asymptotic hyperbola is as follows:

$${\displaystyle \frac{{\left({\sigma }_m-d\right)}^2}{a^2}}-{\displaystyle \frac{J_2}{b^2}}=1$$

(60)

where $d$, $a$, and $b$ are geometric parameters defining the curvature and intercepts of the hyperbola in the meridional plane (Figure 8). By matching the slope and intercept of the hyperbola’s asymptote to those of the MC line, the following relations are obtained:

$${\displaystyle \frac{b}{a}}={\displaystyle \frac{\mathit{sin}\varphi }{K\left(\varphi \right)}},\hspace{1em}d=c\cot \varphi$$

(61)

Substituting Equation (61) into Equation (60), the smoothed yield function is defined as follows:

$$f={\sigma }_m+\sqrt{J_2{K}^2\left(\theta \right)+a^2{\mathit{sin}}^2\varphi }-c\mathit{cos}\varphi =0$$

(62)

This form recovers the original MC surface when $a=0$, while retaining differentiability near the corners. The degree of regularization can be controlled by the parameter $a=0$.

A second limitation of the MC surface concerns its deviatoric-plane representation. At low confining pressures, typical of roadside barrier installations, granular materials do not follow the hexagonal pattern of the MC criterion. Instead, experimental results indicate a triangular deviatoric yield surface at low confinements [96], as illustrated in Figure 9a. The FHWA model incorporates this behavior by modifying the function $K\left(\theta \right)$ following the formulation of Klisinski [101]:

$$K\left(\theta \right)={\displaystyle \frac{4\left(1-e^2\right){\mathit{cos}}^2\theta +{\left(2e-1\right)}^2}{2\left(1-e^2\right)\mathit{cos}\theta +\left(2e-1\right)\sqrt{4\left(1-e^2\right){\mathit{cos}}^2\theta +5e^2-4e}}}$$

(63)

The parameter $e$ defines the ratio of triaxial extension to triaxial compression strength, with $0.5<e\le 1.0$. A value of $e=1.0$ results in a circular, while $e=0.55$ produces a triangular deviatoric surface, Figure 9b, consistent with observed behavior under low confinement.

5.1.3. Flow Rule

The elasto-plastic formulation of the FHWA model assumes the additive decomposition of strain rate into elastic and plastic components. The plastic strain rate tensor is governed by a flow rule of the form:

$${\dot{\epsilon }}_{ij}^{\left(pl\right)}=\dot{\lambda }{\displaystyle \frac{\partial g}{\partial {\sigma }_{ij}}}$$

(64)

where $\dot{\lambda }$ is a plastic multiplier (or consistency parameter), $g$ is the plastic potential, and ${\sigma }_{ij}$ is the stress tensor. The plastic multiplier is subject to the Karush-Kuhn-Tucker conditions [102]:

The consistency condition is provided as follows:

$$\dot{\lambda }\dot{\phi }\left({\sigma }_{ij}\right)=0$$

(65)

For dense granular flows dominated by interparticle friction and collisions, the FHWA model employs an associated flow rule, setting the plastic potential equal to the yield function, $f$. The plastic strain rate becomes the following:

$${\dot{\epsilon }}_{ij}^{\left(pl\right)}=\dot{\lambda }{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}$$

(66)

The consistency parameter is evaluated as follows:

$$\dot{\lambda }={\displaystyle \frac{2G{\dot{\epsilon }}_{ij}\frac{\partial f}{\partial {\sigma }_{ij}}+\left(K-\frac{2G}{3}\right){\dot{\epsilon }}_{kk}\frac{\partial f}{\partial {\sigma }_{ij}}{\delta }_{ij}}{2G\frac{\partial f}{\partial {\sigma }_{pq}}\frac{\partial f}{\partial {\sigma }_{pq}}+\left(K-\frac{2G}{3}\right)\frac{\partial f}{\partial {\sigma }_{pq}}{\delta }_{pq}\frac{\partial f}{\partial {\sigma }_{pq}}{\delta }_{pq}}}$$

(67)

5.1.4. Strain Softening

In dynamic impact scenarios, granular soils surrounding embedded piles display splashing, heaving, and dilation behavior, mimicking a fluid-like response [103,104,105]. To capture this post-yield degradation, the FHWA model integrates a strain energy-based continuum damage formulation [106,107].

The damage initiation criterion, $\xi$, is defined in terms of plastic volumetric strain ${\epsilon }_v^{\left(pl\right)}$ as follows:

$$\xi =-{\displaystyle \frac{1}{K}}\int {\sigma }_{m}d{\epsilon }_v^{\left(pl\right)}$$

(68)

where ${\sigma }_m={\displaystyle \frac{1}{3}}{\dot{\sigma }}_{kk}$ is the mean stress. For dilative response, ${\epsilon }_v^{\left(pl\right)}<0$. The effective stress is reduced by an isotropic damage parameter $D$ as follows:

$${\sigma }_{ij}={\overline{\sigma }}_{ij}\left(1-D\right)$$

(69)

The damage parameter $D$evolves from 0 (undamaged) to 1 (complete failure) and is defined as follows:

$$D={\displaystyle \frac{\xi -{\xi }_0}{\alpha -{\xi }_0}}$$

(70)

where $\alpha$ and ${\xi }_0$ are material parameters denoting the initiation and completion of damage, respectively. The peak-to-residual strength degradation is incorporated through a pressure-dependent maximum damage:

$$D_{max}={\displaystyle \frac{\left(\mathit{sin}{\varphi }_{peak}-\mathit{sin}{\varphi }_{res}\right)}{\mathit{sin}{\varphi }_{peak}}}$$

(71)

To mitigate mesh dependency, the model uses a void formation energy parameter:

$$G_f=V^{{\displaystyle \frac{1}{3}}}\int {\sigma }_{m}d{\epsilon }_v^{\left(pl\right)}$$

(72)

where $V$is the volume of the element.

The strain at full damage, $\alpha$, is computed as follows:

$$\alpha ={\displaystyle \frac{2G_f}{K{\xi }_0{V}^{\frac{1}{3}}}}+{\xi }_0$$

(73)

5.1.5. Viscoplastic Regularization

Granular materials exhibit dual behaviors: solid-like resistance and fluid-like flow, depending on the loading regime. To accommodate this, the FHWA model applies a viscoplastic regularization based on a two-parameter extension of the Duvaut–Lions formulation [108,109,110,111]. This scheme interpolates between the static yield stress ${\overline{\sigma }}_{ij}$ and the elastic trial stress ${\overline{\sigma }}_{ij}^{\left(trial\right)}$ to define viscoplastic stress:

$${\overline{\sigma }}_{ij}^{\left(vp\right)}={\overline{\sigma }}_{ij}\left(1-\varsigma \right)+\varsigma {\overline{\sigma }}_{ij}^{\left(trial\right)}$$

(74)

$$\varsigma ={\displaystyle \frac{\eta }{\Delta t+\eta }}$$

(75)

$$\eta ={\left({\displaystyle \frac{\gamma }{{\dot{\epsilon }}_e}}\right)}^{1-{\displaystyle \frac{1}{n}}}$$

(76)

In which, $\varsigma$ is a stress update parameter, $\Delta t$ is the time step, $\eta$ is the fluidity parameter, $\gamma$ is the viscosity parameter, $n$ is the viscosity exponent, and ${\dot{\epsilon }}_e$ is the effective strain rate which is expressed as follows:

$${\dot{\epsilon }}_e=\sqrt{{\displaystyle \frac{2}{3}}{\dot{e}}_{ij}{\dot{e}}_{ij}}$$

(77)

with deviator strain ${\dot{e}}_{ij}$:

$${\dot{e}}_{ij}={\dot{\epsilon }}_{ij}-{\displaystyle \frac{1}{3}}{\dot{\epsilon }}_{kk}{\delta }_{ij}$$

(78)

As $\eta =0$, the solution approaches the inviscid yield surface, as $\eta \to \infty$, plastic flow is surpassed, and the stress state remains elastic.

5.2. Model Capabilities and Limitations

Table 4. Advantages and limitations of FHWA soil model.

Advantages	Limitations
Captures low confinement behavior in granular backfill	Requires intensive parameter calibration
Incorporates strain-softening and strain rate effects	Requires detailed experimental data
Accurately reflects deviatoric yield shape evolution
Constructed for simulating soil behavior in roadside barrier systems

summarizes the performance characteristics of the FHWA soil model for crash simulation applications.

6. Cap Models

6.1. Fundamentals of Cap Models

Cap models for geomaterials are formulated within the theoretical framework of work-hardening plasticity under the incremental theory, assuming time-independent material behavior. These models distinguish between elastic and plastic strain components during each loading increment. The total strain rate is expressed as the additive decomposition:

$${\dot{\epsilon }}_{ij}={\dot{\epsilon }}_{ij}^{\left(el\right)}+{\dot{\epsilon }}_{ij}^{\left(pl\right)}$$

(79)

Elastic deformations are governed by nonlinear stress-dependent moduli:

$${\dot{\epsilon }}_{ij}^{\left(el\right)}={\displaystyle \frac{I_1}{9K\left({\sigma }_{pq}\right)}}{\delta }_{ij}+{\displaystyle \frac{1}{2G\left({\sigma }_{pq}\right)}}{\dot{s}}_{ij}$$

(80)

where $I_1={\dot{\sigma }}_{kk}$ is the first invariant of the stress tensor, $K\left({\sigma }_{pq}\right)$ and $G\left({\sigma }_{pq}\right)$ are the bulk and shear moduli, respectively, and ${\dot{s}}_{ij}={\dot{\sigma }}_{ij}-{\displaystyle \frac{1}{3}}\left(I_1\right)$ is the deviatoric stress rate, and ${\delta }_{ij}$ is the Kronecker delta. To ensure path independence of the elastic response, shear and bulk moduli are defined as follows [87]:

$$G=G\left(\sqrt{J_2},{\epsilon }_{ij}^{pl}\right)$$

(81)

$$K=K\left(I_1,{\epsilon }_{ij}^{pl}\right)$$

(82)

The plastic strain rate tensor is derived from the plastic potential surface $f$ and follows a non-associated flow rule under admissible loading conditions:

$${\dot{\epsilon }}_{ij}^{pl}=\left\{\begin{array}{c}\dot{\lambda }{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}} if f=0\hspace{0.33em}\mathrm{and}{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{\dot{\sigma }}_{ij}>0\hspace{0.33em}\\ 0 \mathrm{otherwise}\hspace{0.33em}\hfill \end{array}\right.$$

(83)

As illustrated in Figure 10, the loading function $f$ for cap models is composed of two distinct surfaces: (1) a shear failure envelope that defines the limit of deviatoric stress, and (2) a hardening cap surface that evolves with volumetric plastic strain. This composite function is defined as follows:

$$f=\left\{\begin{array}{c}h\left(I_1,\sqrt{J_2}\right)=\sqrt{J_2}-Q\left(I_1\right)\hspace{1em}\mathrm{on} \mathrm{failure} \mathrm{surface}\\ H\left(I_1,\sqrt{J_2},\kappa \right)=\sqrt{J_2}-F\left(I_1,\kappa \right)\hspace{1em}\mathrm{on} \mathrm{cap} \mathrm{surface}\end{array}\right.$$

(84)

The cap evolution is characterized by the hardening parameter $\kappa ,$related to plastic volumetric strain:

$$\kappa =g\left({\epsilon }_{kk}^{pl}\right)$$

(85)

Equation (85) enables both expansion and contraction of the cap, thus allowing control over volumetric dilatancy and preventing unrealistic predictions common in classical Drucker–Prager models.

The plastic stress–strain relations in terms of deviatoric and volumetric components are as follows:

$${\dot{e}}_{ij}^{pl}=\dot{\lambda }{\displaystyle \frac{\partial f}{\partial s_{ij}}}={\displaystyle \frac{\dot{\lambda }}{2\sqrt{J_2}}}{\displaystyle \frac{\partial f}{\partial \sqrt{J_2}}}{s}_{ij}$$

(86)

$${\dot{\epsilon }}_{kk}^{pl}=3\dot{\lambda }{\displaystyle \frac{\partial f}{\partial I_1}}$$

(87)

$${\dot{\epsilon }}_{ij}^{pl}=\dot{\lambda }\left({\displaystyle \frac{1}{2\sqrt{J_2}}}{\displaystyle \frac{\partial f}{\partial \sqrt{J_2}}}{s}_{ij}+{\displaystyle \frac{\partial f}{\partial I_1}}{\delta }_{ij}\right)$$

(88)

The plastic multiplier $\dot{\lambda }$ is determined from the consistency condition:

$$\dot{f}={\displaystyle \frac{\partial f}{\partial I_1}}{\dot{I}}_1+{\displaystyle \frac{1}{2\sqrt{J_2}}}{\displaystyle \frac{\partial f}{\partial \sqrt{J_2}}}{s}_{ij}{\dot{s}}_{ij}+{\displaystyle \frac{\partial f}{\partial \kappa }}{\displaystyle \frac{\partial \kappa }{\partial {\epsilon }_{kk}^{pl}}}{\dot{\epsilon }}_{pp}^{pl}=0$$

(89)

The elastic stress–strain relations are recast in deviatoric and volumetric components as follows:

$${\dot{e}}_{ij}^{el}={\displaystyle \frac{{\dot{s}}_{ij}}{2G\left(\sqrt{J_2},{\epsilon }_{ij}^{pl}\right)}}$$

(90)

$${\dot{\epsilon }}_{kk}^{el}={\displaystyle \frac{{\dot{I}}_1}{3K\left(I_1,{\epsilon }_{ij}^{pl}\right)}}$$

(91)

Using Equations (87), (91), and (92), the consistency condition becomes

$$3K{\dot{\epsilon }}_{kk}^{pl}{\displaystyle \frac{\partial f}{\partial I_1}}+{\displaystyle \frac{G{\dot{e}}_{ij}^{el}}{\sqrt{J_2}}}{\displaystyle \frac{\partial f}{\partial \sqrt{J_2}}}{s}_{ij}+3\dot{\lambda }{\displaystyle \frac{\partial f}{\partial I_1}}{\displaystyle \frac{\partial f}{\partial \kappa }}{\displaystyle \frac{\partial f}{\partial {\epsilon }_{kk}^{pl}}}=0$$

(92)

Substituting the total strain rate from Equation (79), and isolating the elastic and plastic contributions results in the following:

$$3K\left({\dot{\epsilon }}_{kk}-{\dot{\epsilon }}_{kk}^{pl}\right){\displaystyle \frac{\partial f}{\partial I_1}}+{\displaystyle \frac{G}{\sqrt{J_2}}}\left({\dot{e}}_{ij}-{\dot{e}}_{ij}^{pl}\right){\displaystyle \frac{\partial f}{\partial \sqrt{J_2}}}{s}_{ij}+3\dot{\lambda }{\displaystyle \frac{\partial f}{\partial I_1}}{\displaystyle \frac{\partial f}{\partial \kappa }}{\displaystyle \frac{\partial f}{\partial {\epsilon }_{kk}^{pl}}}=0$$

(93)

Finally, inserting the expressions for ${\dot{e}}_{ij}^{pl}$ and ${\dot{\epsilon }}_{kk}^{pl}$ from Equations (86) and (87) into Equation (93) and solving for $\dot{\lambda }$, yields

$$\dot{\lambda }={\displaystyle \frac{3K\frac{\partial f}{\partial I_1}{\dot{\epsilon }}_{kk}+\frac{G}{\sqrt{J_2}}\frac{\partial f}{\partial \sqrt{J_2}}{s}_{ij}{\dot{e}}_{ij}}{9K{\left(\frac{\partial f}{\partial I_1}\right)}^2+G{\left(\frac{\partial f}{\partial \sqrt{J_2}}\right)}^2-3\frac{\partial f}{\partial I_1}\frac{\partial f}{\partial \kappa }\frac{\partial f}{\partial {\epsilon }_{kk}^{pl}}}}$$

(94)

This general form reduces to the elasto-plastic multiplier (Equation (67)) when the loading function does not depend on plastic volumetric strain. Historically, modern cap models evolved from the original two-surface formulations developed by [112,113]. These formulations introduced a composite yield surface composed of a shear failure surface and a hardening cap, with independent evolution laws, enabling accurate modeling of both dilative and compacting behavior in pressure-sensitive geomaterials.

6.2. Geologic Cap Model

The Geologic Cap model in LS-DYNA is formulated in terms of the first and second stress invariants:

$$I_1={\sigma }_1+{\sigma }_2+{\sigma }_3$$

(95)

$$J_2={\displaystyle \frac{1}{6}}\left[{\left({\sigma }_1-{\sigma }_3\right)}^2+{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2\right]$$

(96)

where ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$ and compressive stresses are taken as negative.

This two-invariant model describes elastic deformation, shear failure, and cap plasticity through an associated, rate-independent plasticity formulation. The yield surface consists of two intersecting components: a shear failure envelope and a strain-hardening cap. When the stress state lies within the composite surface, the material response is purely elastic. The shear failure surface is defined as follows:

$$F_s=\sqrt{J_2}-\alpha +\gamma \exp \left(-\beta I_1\right)-\theta I_1$$

(97)

where $\alpha$, $\gamma$, $\beta$, and $\theta$ are parameters calibrated to the experimental response. Plastic deformation on the shear surface includes both deviatoric and dilatational components.

The cap surface accounts for irreversible volumetric compaction under non-hydrostatic compressive states. It takes an elliptical form:

$$\sqrt{J_2}={\displaystyle \frac{1}{R}}\sqrt{{\left[X\left(\kappa \right)-L\left(\kappa \right)\right]}^2-{\left[I_1-L\left(\kappa \right)\right]}^2}$$

(98)

where $R$ controls the ellipticity, $\kappa$ is a hardening parameter and, $X\left(\kappa \right), L\left(\kappa \right)$ determine the cap’s location in stress space. These are defined as follows:

The relation between the cap surface shape parameters is provided by the following:

$$X\left(\kappa \right)=L\left(\kappa \right)+R\left[\alpha -\gamma \exp \left(-\beta L\left(\kappa \right)\right)+\theta L\left(\kappa \right)\right]$$

(99)

$$L\left(\kappa \right)\equiv \left\{\begin{array}{c}\kappa \hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}\kappa <0\hspace{1em}\\ 0\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}\kappa \ge 0\hfill \end{array}\right.$$

(100)

The evolution of the cap is governed by a hardening law tied to the plastic volumetric strain rate, defined as follows:

$${\overline{\dot{\epsilon }}}_{kk}^{pl}=W\left[\exp \left\{D_1\left(X-X_O\right)-D_2{\left(X-X_O\right)}^2\right\}-1\right]$$

(101)

where $W$, $D_1$, and $D_2$ are material constants, $X_0$ denotes the initial cap location. The cap must intersect the shear surface with horizontal tangency, ensuring that the shear envelope represents the critical state line separating compactive from dilative volumetric plastic flow. ${\overline{\dot{\epsilon }}}_{kk}^{pl}$ is a history-dependent functional of ${\dot{\epsilon }}_{kk}^{pl}$ defined as follows:

$${\overline{\dot{\epsilon }}}_{kk}^{pl}\equiv min\left({\dot{\epsilon }}_{kk}^{pl},0\right)$$

(102)

Despite its theoretical elegance, the Geologic Cap Model exhibits limitations. The flow direction is not uniquely defined at the cap–shear intersection. Pre-failure dilation is not permitted, contradicting experimental results. Additionally, the numerical requirement of finding a horizontal tangent intersection between the cap and the shear surface incurs significant computational cost. These issues have motivated enhanced formulations, most notably the smooth cap model developed by Schwer and Murray [114,115].

6.3. Schwer Murray Cap Model

The Schwer Murray Cap model is a three-invariant extension of the geologic cap model. It eliminates the discontinuities at the cap–shear interface through a smooth yield surface construction. The model is also referred to as the Continuous Surface Cap model (CSCM) [73]. Figure 11 shows the general shape of the yield surface in both three and two dimensions. To ensure continuity between the cap and shear surfaces, the model utilizes the Pelessone smoothing function [116]. The resulting formulation accounts for isotropic hardening, kinematic shear hardening, and Lode angle dependence. It also accommodates strain rate effects and strain-softening through continuum damage mechanics [111,112,113,114,115,116,117,118].

The general form of the yield function is as follows:

$$\Gamma \left(I_1,J_2,J_3,\kappa ,{\alpha }_{ij}\right)={\Gamma }^2\left(\theta \right)\left(J_2-J_2^{\alpha }\right)-{\left(F_s-N\right)}^2{F}_c$$

(103)

where ${\alpha }_{ij}$ is the back-stress tensor, and $J_2^{\alpha }$ is the scalar measure of kinematic hardening:

$$J_2^{\alpha }={\alpha }_{ij}\left(s_{ij}-{\displaystyle \frac{{\alpha }_{ij}}{2}}\right)$$

(104)

The Lode angle dependence is introduced through the following function:

$$\Gamma \left(\theta \right)={\displaystyle \frac{1}{2}}\left\{\left[1+\mathit{sin}\left(3\theta \right)\right]+{\displaystyle \frac{1}{\psi }}\left[1-\mathit{sin}\left(3\theta \right)\right]\right\}$$

(105)

where $\psi$ is the ratio of the yield stress in triaxial extension to the yield stress in triaxial compression, and the lode angle $\theta$ is defined as follows:

$$\begin{gathered} \theta ={\displaystyle \frac{1}{3}}a\mathit{sin}\left(-{\displaystyle \frac{27}{2}}{\displaystyle \frac{J_3}{{\left(3J_2\right)}^{\frac{3}{2}}}}\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}for\hspace{0.33em}-{\displaystyle \frac{\pi }{6}}\le \theta \le {\displaystyle \frac{\pi }{6}}\Gamma \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{2}}\left\{\left[1+\mathit{sin}\left(3\theta \right)\right]+{\displaystyle \frac{1}{\psi }}\left[1-\mathit{sin}\left(3\theta \right)\right]\right\} \end{gathered}$$

(106)

and the third invariant of deviatoric is as follows:

$$J_3={\displaystyle \frac{1}{3}}{s}_{ij}{s}_{jk}{s}_{ki}$$

(107)

The Pelessone smoothing function used to define the cap region is as follows:

$$F_c\left(I_1,\kappa \right)=1+{\displaystyle \frac{\left[I_1-L\left(\kappa \right)\right]\left[\left|I_1-L\left(\kappa \right)\right|-\left(I_1-L\left(\kappa \right)\right)\right]}{2{\left[X\left(\kappa \right)-L\left(\kappa \right)\right]}^2}}$$

(108)

The functional form for $X\left(\kappa \right)$ and $L\left(\kappa \right)$ are preserved as follows:

$$X\left(\kappa \right)=L\left(\kappa \right)+R\left[\alpha -\gamma \exp \left(-\beta L\left(\kappa \right)\right)+\theta L\left(\kappa \right)\right]$$

(109)

$$L\left(\kappa \right)\equiv \left\{\begin{array}{c}\kappa \hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}\kappa <0\hspace{1em}\\ 0\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}\kappa \ge 0\hfill \end{array}\right.$$

(110)

$L\left(\kappa \right)\le I_1\le X\left(\kappa \right)$ and unity for $I_1<L\left(\kappa \right)$, ensuring a smooth yield surface.

Strain rate effects are incorporated via a modified Duvaut–Lions viscoplastic framework, where the dynamic stress state interpolates between the elastic trial stress and the quasi-static yield surface. The model also includes a continuum damage formulation to capture strain-softening under dynamic loading. Implementation details are provided in [111,112,113,114,115,116,117,118].

6.4. Advantages and Limitations of Cap Soil Models

The advantages and limitations of cap models, i.e., Geologic Cap and Schwer Murray Cap for modeling dynamic soil–structure interaction processes, are summarized in

Table 5. Advantages and limitations of cap soil models.

Model	Advantages	Limitations
Geologic Cap	Can simulate low confinement effects, which is critical in most roadside safety applications	Cannot predict moisture or pore-pressure build-up during undrained failure under dynamic shear loading
	Satisfies the theoretical requirements of uniqueness, stability, and continuity	Does not account for strain-softening or damage and strain rate effects
	Provides proper control of dilatancy	Two-invariant formulation; cannot simulate lower strength in triaxial extension
		Non-smooth yield surface behavior at very low shear stresses causes the algorithm to be complex and inefficient
Schwer Murray Cap	Accounts for strain-softening or damage and strain rate effects	Relatively complicated model with more than 25 input parameters
	Can simulate low confinement effects, which is critical in roadside safety applications	Relatively complicated model with more than 25 input parameters
	Three-invariant formulation; can simulate lower strength in triaxial extension	Cannot predict moisture or pore-pressure build-up during undrained failure under dynamic shear loading
	Smooth yield surface behavior at very low shear stresses; suitable for finite element implementations

.

7. Nested Surface Models

Nested surface models introduce multiple yield surfaces within the stress space of an element to represent material behavior across different stress levels. In these models, yield surfaces are fixed in stress space and do not translate in response to loading, thus eliminating the need for kinematic hardening rules [119]. This approach is particularly advantageous for cyclic loading simulations, as it captures hysteretic energy dissipation without introducing internal variables for surface motion.

Over the past two decades, nested surface formulations have been employed as viable alternatives to isotropic hardening models for simulating the cyclic response of geomaterial [120]. Within the LS-DYNA simulation environment, two nested surface models are available: Hysteretic Soil and Soil Brick. Only the former is applicable to granular soils typically encountered in roadside safety systems, and thus is the focus of this discussion.

7.1. Hysteretic Soil Model

The Hysteretic Soil model is a three-dimensional, piecewise nonlinear, nested surface formulation developed for simulating the inelastic cyclic response of granular soils [73]. It allows for up to ten superposed elastic-perfectly plastic layers, each characterized by distinct elastic moduli and yield thresholds. Hysteretic energy dissipation arises as individual layers yield under increasing deviatoric strain amplitude. The overall deviatoric stress is computed as the sum of contributions from each layer.

The conceptual basis of the model originates from Iwan’s one-dimensional slider-spring system [121], later extended by Chiang and Beck [119] to three dimensions using a von Mises yield criterion and an associated flow rule. The layered structure and the resulting cyclic and monotonic response of a typical four-layer configuration are shown in Figure 12.

In this model, each layer is represented by an elastic-perfectly plastic deviatoric response. The total deviatoric stress tensor is obtained by summing the stress tensors of all layers:

$$f\left(J_2\right)={\tau }_y$$

(111)

where $J_2$ is the second invariant of the deviatoric stress tensor, and ${\tau }_y$ is shear yield stress for a given layer. The model supports up to ten layers, limiting the discretization of the shear stress–strain backbone curve to ten points

To incorporate pressure-dependent yield behavior, a modified Drucker–Prager criterion is available:

$${\tau }_y\left(p\right)=\sqrt{{\displaystyle \frac{a_0+a_1\left(p-p_0\right)+a_2{\left(p-p_0\right)}^2}{a_0+a_1{p}_{ref}+a_2{p}_{ref}^2}}}{\tau }_y\left(p_{ref}\right)$$

(112)

Here, $p$ is the hydrostatic (mean effective) stress, $a_0$, $a_1$, and $a_2$are curve fitting parameters, $p_0$ is the cut-off pressure below which failure occurs, and $p_{ref}$ is the reference pressure. Putting $a_0=0$ $a_1=0$, and $a_2=0$ results in the classical Drucker–Prager model.

Elastic stiffness moduli are also pressure-dependent and follow a power-law form [123]:

$$G\left(p\right)={\left({\displaystyle \frac{p-p_0}{p_{ref}}}\right)}^{b}G\left(p_{ref}\right)$$

(113)

$$K\left(p\right)={\left({\displaystyle \frac{p-p_0}{p_{ref}}}\right)}^{b}K\left(p_{ref}\right)$$

(114)

where $G\left(p\right)$ and $K\left(p\right)$ are the shear and bulk moduli at confining pressure $p$, $G\left(p_{ref}\right)$ and $K\left(p_{ref}\right)$ are the corresponding moduli at reference pressure, and $b$ is the exponent controlling the pressure sensitivity. This formulation enables the simulation of nonlinear compressibility and shear stiffness evolution with confinement. The model also supports user-defined input for dilatancy and strain rate sensitivity, essential for dynamic loading conditions involving high strain rates and localized volumetric expansion. Details of these features are described in [73].

7.2. Soil Brick Model

The Soil Brick model is a kinematic yield surface formulation implemented in strain space. It accounts for anisotropy and is tailored to model the behavior of over-consolidated clays [124]. The model is not suitable for simulating granular soils, and therefore has been excluded from this evaluation focused on roadside safety systems involving pile–granular soil interactions. Further theoretical details and input parameter specifications are available in [123,125,126].

7.3. Advantages and Limitations of Nested Surface Soil Models

One of the advantages of the Hysteretic Soil model is the soil testing data in the form of shear stress and strains can be utilized to estimate most of the required material constants. The advantages and limitations of the Hysteretic Soil for roadside safety applications, including large deformation computational modeling of laterally impacted pile-granular soil systems, are summarized in

Table 6. Advantages and limitations of the Hysteretic Soil model.

Advantages	Limitations
Can reproduce the hysteretic response within the failure surface	Cannot simulate moisture or pore-pressure build-up during undrained dynamic shear loading
Accounts for dilatancy and strain rate effects	Does not incorporate strain-softening or continuum damage
	Cannot accurately simulate low-confinement behavior, which is critical in many roadside safety applications
	Requires extensive soil characterization to calibrate more than 20 input parameters

.

8. Evaluation of Soil Constitutive Models

A theoretical evaluation was conducted on the soil constitutive models implemented in the LS-DYNA simulation platform. This evaluation focused on identifying the fundamental characteristics of each model and assessing its applicability to the simulation of granular soil behavior under conditions representative of roadside safety applications. The models were classified based on their constitutive structure, including yield surface formulation, flow rule, hardening behavior, and the ability to reproduce key aspects of granular soil response.

The evaluation was performed to support the selection of appropriate models for numerical analysis of soil-embedded structural systems subjected to vehicle impact. The classification is based on the extent to which each model incorporates mechanisms required to simulate realistic soil behavior under high-frequency dynamic loading. The models were organized into three categories, named to reflect their constitutive completeness and their relevance to high-fidelity dynamic soil–structure interaction simulations.

8.1. Comprehensive Constitutive Models

These models incorporate the principal features required to simulate granular soil under dynamic loading. Their formulations are fully three-dimensional and include mechanisms to capture low-confinement response, volumetric dilation during shear, and strain-softening with damage evolution. Regularization is employed to preserve the well-posedness of the governing equations in softening regimes. These models also include rate sensitivity and can simulate moisture or pore-pressure accumulation under undrained conditions. The yield surface is expressed using all three stress invariants, enabling the model to differentiate between behavior in triaxial compression and triaxial extension.

8.2. Intermediate Capability Models

Models in this category exhibit several of the above features but omit one or more of the critical elements required for complete representation of granular soil behavior. Examples include models that incorporate pressure-sensitive plasticity or cyclic loading capabilities but lack strain rate effects, softening behavior, or moisture–pore pressure coupling. These models are suitable for specific applications but may require simplifications when used outside their intended scope.

8.3. Foundational Yield Models

These models are based on early plasticity theories and do not include softening, dilation, or rate effects. The yield surfaces are typically defined using two stress invariants, and there is no mechanism to account for pore-pressure generation or post-yield volumetric strain. While these models may be used for preliminary or idealized simulations, they are not suitable for predicting the complex response of granular soil under lateral impact.

The classification reflects the theoretical formulation and numerical behavior of each model as implemented in LS-DYNA. The group assignments represent the authors’ interpretation of each model’s capability to simulate granular soil response in impact scenarios. The attributes and corresponding group designations are summarized in

Table 7. LS-DYNA soil constitutive models, their attributes, and theoretical formulations.

Type of Soil Model	Soil Model	Type of Soil			Failure Surface	Flow Rule: Associated or Non-associated	Hardening Criteria
		Granular	Cohesive	Rock
Elastic	Elastic	Yes	Yes	Yes	NA	NA	NA
	Biot Hysteretic	Yes	Yes	Yes	NA	NA	NA
Simple Elastoplastic	Soil and Foam	Yes	Yes	Yes	Extended von Mises	Associated	None
	Soil and Foam Failure	Yes	Yes	Yes	Extended von Mises	Associated	None
	Pseudo Tensor	Yes	Yes	Yes	Extended von Mises	Associated	None
Elastoplastic	Soil Concrete	Yes	Yes	Yes	Extended von Mises	Associated	None
	Mohr–Coulomb	Yes	Yes	Yes	Mohr–Coulomb	Non-associated	None
	Drucker–Prager	Yes	Yes	Yes	Extended von Mises	Non-associated	None
	Jointed Rock	No	No	Yes	Mohr–Coulomb	Non-associated	None
Elasto-viscoplastic	FHWA	Yes	Yes	Yes	Modified Mohr–Coulomb	Associated	Isotropic
Cap	Geologic Cap	Yes	Yes	Yes	Curved extended von Mises	Associated	Isotropic
	Schwer Murray Cap	Yes	Yes	Yes	Curved extended von Mises	Non-associated	Kinematic
Nested surface	Hysteretic	Yes	Yes	Yes	Extended von Mises	Non-associated	Nested surfaces
	Soil Brick	No	Yes	No	Extended von Mises	Associated	Nested surfaces

and

Table 8. Evaluation of LS-DYNA soil constitutive models for simulating granular soil behavior under impact loading.

Soil Model	Does Model Include/Predict						Model Group
	Three-Dimensional Behavior	Low Confinement Effects	Dilation	Strain Softening	Strain rate Effects	Moisture/Pore pressure
Elastic	Yes	No	No	No	No	No	Foundational Yield Model
Biot Hysteretic	Yes	No	No	No	No	No	Foundational Yield Model
Soil and Foam	Yes	No	No	No	No	No	Foundational Yield Model
Soil and Foam Failure	Yes	No	No	No	No	No	Foundational Yield Model
Pseudo Tensor	Yes	No	No	Yes	Yes	No	Intermediate Capability Model
Soil Concrete	Yes	No	No	Yes	No	No	Foundational Yield Model
Mohr–Coulomb	Yes	No	Yes	No	No	No	Foundational Yield Model
Drucker–Prager	Yes	No	Yes	No	No	No	Foundational Yield Model
Jointed Rock	Yes	No	Yes	No	No	No	Foundational Yield Model
FHWA	Yes	Yes	Yes	Yes	Yes	Yes	Comprehensive Constitutive Model
Geologic Cap	Yes	Yes	Yes	No	No	No	Intermediate Capability Model
Schwer Murray Cap	Yes	Yes	Yes	Yes	Yes	No	Intermediate Capability Model
Hysteretic Soil	Yes	No	Yes	No	Yes	No	Intermediate Capability Model
Soil Brick	Yes	No	No	No	No	No	Foundational Yield Model

.

9. Summary and Future Research Directions

The selection of appropriate soil constitutive models for simulating dynamic soil–structure interaction processes remains a critical challenge in computational geomechanics. Granular soil response under impact loading is highly dependent on confinement, packing fraction, strain rate, moisture condition, and post-yield behavior. Accurate prediction of these effects requires models that can represent nonlinear elasticity, plasticity, dilation, strain softening, and rate sensitivity under three-dimensional loading conditions.

This work presented a comprehensive theoretical evaluation of soil constitutive models available in LS-DYNA, with emphasis on their capability to simulate granular soil behavior under dynamic impact loading. Models were categorized based on their formulation type, yield surface structure, flow rule, and hardening mechanism. While simple elasto-plastic and foundational models are easy to implement, they provide limited insight into high-rate deformation and moisture-related effects. More advanced models, including cap and viscoplastic formulations, incorporate features required to simulate complex stress paths, but often involve numerous input parameters and increased computational demand.

Conventional finite element methods remain the primary tool for simulating soil–structure interaction processes under impact loading. However, they face limitations when applied to problems involving large deformation and strain localization. Coupling with regularization methods and multiphysics models, such as those accounting for pore pressure and unsaturated behavior, is needed for improved predictive accuracy. Additionally, the implementation of three-invariant yield surfaces is essential for capturing differences in response under triaxial compression and extension.

Foundational yield models may be adequate for idealized or preliminary simulations, such as assessing elastic–plastic soil behavior under simplified boundary conditions or conducting sensitivity analyses with coarse soil parameter estimates. Intermediate capability models are appropriate for simulations involving moderate confinement or cyclic loading conditions. Comprehensive constitutive models, including the FHWA and Schwer Murray Cap models, are necessary for high-fidelity crash simulations of dynamic soil-structure interaction processes, where low-confinement dilation, strain softening, and strain rate effects govern the dynamic response of granular soil.

Future research should focus on validating model predictions against full-scale crash data and laboratory tests under controlled confinement and strain rate conditions. Parametric studies are needed to reduce uncertainty in input parameters and assess the influence of soil variability. Integration with mesh-free or particle-based approaches may also be explored to overcome current limitations in modeling discontinuities and fluid–solid transition in granular materials during impact. This study provides a framework for selecting and classifying LS-DYNA soil models for impact problems involving granular soils and identifies areas where further model refinement and experimental validation are necessary.